Problem: What is the number of sides in a regular polygon when the measure of each interior angle is $162^\circ$?
Answer: We can use the fact that the exterior angles of any polygon sum to $360^\circ$. Each exterior angle (one at each vertex) will be supplementary to an interior angle, since an exterior angle is the angle between a side and the extension of an adjacent side. Therefore, each exterior angle equals $180-162=18^\circ$, so there are $360\div 18=\boxed{20}\text{ sides}$.